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/* ndtri.c
*
* Inverse of Normal distribution function
*
*
*
* SYNOPSIS:
*
* double x, y, ndtri();
*
* x = ndtri( y );
*
*
*
* DESCRIPTION:
*
* Returns the argument, x, for which the area under the
* Gaussian probability density function (integrated from
* minus infinity to x) is equal to y.
*
*
* For small arguments 0 < y < exp(-2), the program computes
* z = sqrt( -2.0 * log(y) ); then the approximation is
* x = z - log(z)/z - (1/z) P(1/z) / Q(1/z).
* There are two rational functions P/Q, one for 0 < y < exp(-32)
* and the other for y up to exp(-2). For larger arguments,
* w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0.125, 1 20000 7.2e-16 1.3e-16
* IEEE 3e-308, 0.135 50000 4.6e-16 9.8e-17
*
*
* ERROR MESSAGES:
*
* message condition value returned
* ndtri domain x <= 0 -NPY_INFINITY
* ndtri domain x >= 1 NPY_INFINITY
*
*/
/*
* Cephes Math Library Release 2.1: January, 1989
* Copyright 1984, 1987, 1989 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#include "mconf.h"
/* sqrt(2pi) */
static double s2pi = 2.50662827463100050242E0;
/* approximation for 0 <= |y - 0.5| <= 3/8 */
static double P0[5] = {
-5.99633501014107895267E1,
9.80010754185999661536E1,
-5.66762857469070293439E1,
1.39312609387279679503E1,
-1.23916583867381258016E0,
};
static double Q0[8] = {
/* 1.00000000000000000000E0, */
1.95448858338141759834E0,
4.67627912898881538453E0,
8.63602421390890590575E1,
-2.25462687854119370527E2,
2.00260212380060660359E2,
-8.20372256168333339912E1,
1.59056225126211695515E1,
-1.18331621121330003142E0,
};
/* Approximation for interval z = sqrt(-2 log y ) between 2 and 8
* i.e., y between exp(-2) = .135 and exp(-32) = 1.27e-14.
*/
static double P1[9] = {
4.05544892305962419923E0,
3.15251094599893866154E1,
5.71628192246421288162E1,
4.40805073893200834700E1,
1.46849561928858024014E1,
2.18663306850790267539E0,
-1.40256079171354495875E-1,
-3.50424626827848203418E-2,
-8.57456785154685413611E-4,
};
static double Q1[8] = {
/* 1.00000000000000000000E0, */
1.57799883256466749731E1,
4.53907635128879210584E1,
4.13172038254672030440E1,
1.50425385692907503408E1,
2.50464946208309415979E0,
-1.42182922854787788574E-1,
-3.80806407691578277194E-2,
-9.33259480895457427372E-4,
};
/* Approximation for interval z = sqrt(-2 log y ) between 8 and 64
* i.e., y between exp(-32) = 1.27e-14 and exp(-2048) = 3.67e-890.
*/
static double P2[9] = {
3.23774891776946035970E0,
6.91522889068984211695E0,
3.93881025292474443415E0,
1.33303460815807542389E0,
2.01485389549179081538E-1,
1.23716634817820021358E-2,
3.01581553508235416007E-4,
2.65806974686737550832E-6,
6.23974539184983293730E-9,
};
static double Q2[8] = {
/* 1.00000000000000000000E0, */
6.02427039364742014255E0,
3.67983563856160859403E0,
1.37702099489081330271E0,
2.16236993594496635890E-1,
1.34204006088543189037E-2,
3.28014464682127739104E-4,
2.89247864745380683936E-6,
6.79019408009981274425E-9,
};
double ndtri(y0)
double y0;
{
double x, y, z, y2, x0, x1;
int code;
if (y0 <= 0.0) {
mtherr("ndtri", DOMAIN);
return (-NPY_INFINITY);
}
if (y0 >= 1.0) {
mtherr("ndtri", DOMAIN);
return (NPY_INFINITY);
}
code = 1;
y = y0;
if (y > (1.0 - 0.13533528323661269189)) { /* 0.135... = exp(-2) */
y = 1.0 - y;
code = 0;
}
if (y > 0.13533528323661269189) {
y = y - 0.5;
y2 = y * y;
x = y + y * (y2 * polevl(y2, P0, 4) / p1evl(y2, Q0, 8));
x = x * s2pi;
return (x);
}
x = sqrt(-2.0 * log(y));
x0 = x - log(x) / x;
z = 1.0 / x;
if (x < 8.0) /* y > exp(-32) = 1.2664165549e-14 */
x1 = z * polevl(z, P1, 8) / p1evl(z, Q1, 8);
else
x1 = z * polevl(z, P2, 8) / p1evl(z, Q2, 8);
x = x0 - x1;
if (code != 0)
x = -x;
return (x);
}
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